Proving that multiplication of convex function is convex

Prove/disprove the next statement:
Let f,g two convex functions, then h(x)=f(x)⋅g(x) is also convex


So, we know that h ′ (x)=f ′ (x)⋅g(x)+f(x)⋅g ′ (x) . We also know that f ′ (x),g ′ (x) are monotonically increasing because they are convex. If I can show that h ′ (x) is also monotonically increasin I'm done, but I'm not sure how to do it. Any hints?

Hint: You can write the function defined by x↦−x 2 as the product of two very simple linear and hence convex functions.

Ref. http://math.stackexchange.com/questions/147664/proving-that-multiplication-of-convex-function-is-convex

Sum of two convex functions is convex.

Show that sum of two convex functions is convex.?

Answer:

Consider 2 convex functions g(x) and f(x).
Using property of second derivative test we have:
g(x)'' > 0 and f(x)'' > 0.
We are interested in showing that y(x) = g(x) + f(x) is also convex.
y(x)'' = c*g(x)'' + k*f(x)'', where c and k are positive numbers (where is no convex function those coeficients after diferentiation will give you negative numbers).
Because c, g(x)'', k, f(x)'' are > 0 , we get y(x)'' also >0, hence y(x) is a convex function.

Ref. http://wiki.answers.com/Q/Show_that_sum_of_two_convex_functions_is_convex.